Spectral collocation method coupled with domain decomposition for exterior problems of the Fisher equation

The spectral collocation method coupled with domain decomposition is developed for solving the exterior problems of the Fisher's equation with polygon obstacles. Some results on the composite Laguerre-Legendre interpolation, which is a set of piecewise mixed interpolations coupled with domain decomposition, are introduced. As an important application, the composite spectral collocation scheme based on the Legendre-Gauss-Lobatto and the Laguerre-Gauss-Radau nodes is provided for the exterior problems of the Fisher's equation. The convergence of the proposed scheme is proved. Efficient algorithm is implemented. Numerical results demonstrate the high accuracy in space of the proposed method and confirm the theoretical analysis well. The approximation results and some techniques developed in this paper are also very useful for other exterior nonlinear problems with complex geometry.